STAT1301 T10 Solution

STAT1301 T10 Solution - THE UNIVERSITY OF HONG KONG...

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Unformatted text preview: THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1301 PROBABILITY AND STATISTICS I EXAMPLE CLASS 10 EX.1 Let X and Y be random variables with joint pdf my, if0<zv<y<oo f(:c,y)={ y 0, otherwise Let W=Y~X, Z=Y+X. Find thejoint pdfofT/V and Z. uz7~x,%=7+>< ; “2’-” “fig 0 Z ,— C) X”:— , T, L {fa/L5wa J H) JIYIXfi): %% .:/’7”l§ iii/:11 %%~%§g : Z Z _ {2%) 2%)8 fix”): 1:5):- {31/ 2.. — wig—U— EX.2 Suppose X and Y have joint density function flay) ={ (a) Find the joint density function of U :z X Y and V = X / Y. (b) Find the marginal pde of U and V. I 1 $2712 7 0, fim>Ly>1 otherwise A) U=X7’ , VL—é ® X=fi/7, 7:]; (XM 7 mix )2 posifidd) Hence, 1% “bu/(WM 25 l—l. fl 2.9: as [K EA. Joey)?“ 3" am 2” 2—4-1.-1 g2 % gin *M 41/ ¥V«.2V 6U -& 3W 6% jam « « M‘FéZ/MM 1/0qu by i 7, -1: -.__l__\ I 7A,V@W) bu/M 2V/“2m/ ) ZiSI/SM,/ ,U‘f 1’) Mufiifimw : J L‘ 2w[/&VJ—Lg .. fix 7 / KM<°° fiSng‘ é?) MamflxGé/V) 5r 0<V<( , - “Q! i I Id», _ 10”" 527174“ 2:7 72' .L ~ 2/ V ’m, s< 6 41/“) 064,61 ,.L -1“:J. V”): Law “21/ M r 2% EX.3 Suppose X, Y and Z are independently and identically distributed as E$p(1). Derive the joint distribution of U 2 X + Y, V = X + Z and W = Y + Z. flag“ MM% King a 10020253): 57‘ 9'1 52 : €~c9aflfj I 27aykaz70 U= X”, W X+g , N: we A k ( Q7 )<- Emma) / rid/mm) , %=:’(V+N~U) Hem ?f Z5 A H mflmfim BJA 9M 9a ( I Jzatx-é-Bfi O" 9Kflav2(0(:‘l 9x395? engage 9’{ exaggfi [AL govt We (6 U, mu :; 5c"ij -1 “.1 stud/IN) : 6 20/06 VfLfl/KJ/ : 204+W“)/[w/0J<Mfl/, 1/20,“),0 L ,4; 7670 I WWW?) .u<mV C 370 é/[l/th/‘ng @{LwV—M <1? W~VI<H<WVJ VfLH/QD N ? WV \ EX.4 __ . Suppose X, Y 1131 U(0,1). Find the pdf of Z = X + Y. 7;; dbfrlbafim {‘M/(Cflznofi :Z (’5 diam @L 6%) 2mg): P(X+“rga~/ MW Oézq, g Hf flu“ 5595/» fa {(193)5ng : £%[9%’7(( K jfif 2. UM ./<Tzs2, _ m“ @592 ff my)an K1275; Z flforflmflédfl‘ r W MW 14/??? W 3 (7:4) 7‘ [w’gjéfl > 2%~§ /( 6/1 H ca 1 O r K?) GWZXEE , osau 2%”354 r [<2—i2 { r %}l 7W 12 p665 [3 {W by ' :6 , 0S£<f 4 {4?}: {25% r ( (E52 v > ' mag, ...
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STAT1301 T10 Solution - THE UNIVERSITY OF HONG KONG...

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